Abstract
Metal-organic frameworks (MOFs) are nanoporous materials with good prospects as recognition elements for gas sensors owing to their adsorptive sensitivity and selectivity. A gravimetric, MOF-based sensor functions by measuring the mass of gas adsorbed in a MOF. Changes in the gas composition are expected to produce detectable changes in the mass of gas adsorbed in the MOF. In practical settings, multiple components of the gas adsorb into the MOF and contribute to the sensor response. As a result, there are typically many distinct gas compositions that produce the same single-sensor response. The response vector of a gas sensor array places multiple constraints on the gas composition. Still, if the number of degrees of freedom in the gas composition is greater than the number of MOFs in the sensor array, the map from gas compositions to response vectors will be non-injective (many-to-one). Here, we outline a mathematical method to determine undetectable changes in gas composition to which non-injective gas sensor arrays are unresponsive. This is important for understanding their limitations and vulnerabilities. We focus on gravimetric, MOF-based gas sensor arrays. Our method relies on a mixed-gas adsorption model in the MOFs comprising the sensor array, which gives the mass of gas adsorbed in each MOF as a function of the gas composition. The singular value decomposition of the Jacobian matrix of the adsorption model uncovers (i) the unresponsive directions and (ii) the responsive directions, ranked by sensitivity, in gas composition space. We illustrate the identification of unresponsive subspaces and ranked responsive directions for gas sensor arrays based on Co-MOF-74 and HKUST-1 aimed at quantitative sensing of CH4/N2/CO2/C2H6 mixtures relevant to natural gas sensing.
Highlights
Gas sensors [1] have a wide range of applications in the chemical industry and in emerging domains such as air quality monitoring [2], diagnosis of disease [3], food quality assessment [4], detection of chemical warfare agents and explosives [5, 6], and crop monitoring [7].Metal-organic frameworks (MOFs) [8] are nanoporous materials with high prospects as recognition elements for enhanced gas sensors [9,10,11,12] owing to their sensitivity and selectivity
Our work focuses on identifying the limitations of non-injective, MOF-based gas sensor arrays by elucidating directions in composition space in which the gas sensor is unresponsive to changes
The inverse problem [35] of predicting the gas composition x from the response vector m of a non-injective gas sensor array typically has infinite solutions, meaning that infinitely many gas compositions are consistent with the response
Summary
Gas sensors [1] have a wide range of applications in the chemical industry and in emerging domains such as air quality monitoring [2], diagnosis of disease [3], food quality assessment [4], detection of chemical warfare agents and explosives [5, 6], and crop monitoring [7]. Cross-sensitivity plagues gas sensors in Figure 1: A gravimetric sensor comprised of a thin film of metal-organic framework (MOF) [23] (the recognition element), in this case HKUST-1 [32], mounted on a quartz crystal microbalance (QCM) [22,33] (the signal transducer). Unresponsive directions follow from the null space of the Jacobian We illustrate this under the context of quantitative sensing for diluents and higher-hydrocarbons in natural gas [45] (N2, CO2, and C2H6 in CH4) using gravimetric, MOF-based sensors comprised of HKUST-1 and Co-MOF-74. Our work focuses on identifying the limitations of non-injective, MOF-based gas sensor arrays by elucidating directions in composition space in which the gas sensor is unresponsive to changes
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